Proof of Bernoulli's Law

Date: 6/22/96 at 15:29:33
From: peter
Subject: Probability - Bernoulli's law of large numbers
I am a high school student from Sweden and I would like to know how
one can prove Bernoulli's law of large numbers, which states that if
an experiment is repeated n times, the relative frequencey of an
occurrence tends to a fixed number "p" as n tends to infinity.
This theorem seems to me to be more of an empirical fact than a
derived theorem of axiomatic math, so I really wonder...
Thank you.

Date: 6/23/96 at 20:42:20
From: Doctor Anthony
Subject: Re: Probability - Bernoulli's law of large numbers
Explanation of Law of Large Numbers.
Let X(r) be a sequence of mutually independent random variables with a
common distribution. If the expectation MU = E(X(r)) exists then:
PR[{(X(1)+X(2)+..+X(n)}/n - MU > k] -> 0 as n -> infinity for any k>0
If S(n) = X(1)+X(2)+....+X(n) and if the variance is VAR(X) then by
Chebyshev's inequality:
PR[S(n)>t] <or= nVAR(X)/t^2
For t = kn the right hand side tends to 0 and so:
PR[S(n)>kn] -> 0
PR[S(n)/n > k] -> 0
PR[S(n)/n - MU > k] -> 0
Which is the condition we wished to prove.
-Doctor Anthony, The Math Forum
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